Parallelogram Side Length: Finding AC When Perimeter = 21 and AB = 7

Q: Why don't we add all four sides of the parallelogram?

Great question! In a parallelogram, opposite sides are equal. So if AB = 7, then CD = 7 too. If AC = 0.5X, then BD = 0.5X. The formula P = 2(side₁ + side₂) automatically accounts for both pairs!

Q: How do I know which sides are adjacent in the parallelogram?

Look at the diagram! Adjacent sides share a common vertex (corner). In this parallelogram, AB and AC share vertex A, so they're adjacent. Use these two different side lengths in your formula.

Q: What if I get a different value for X?

Double-check your algebra! Start with $ 21 = 2(7 + 0.5X) $, then divide by 2: $ 10.5 = 7 + 0.5X $. Subtract 7: $ 3.5 = 0.5X $. Finally, divide by 0.5: $ X = 7 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find AC

00:03 Opposite sides are equal in parallelograms

00:10 They are also a pair of opposite sides therefore equal

00:21 The perimeter of the parallelogram equals the sum of its sides

00:35 Let's substitute appropriate values and solve for X

00:51 Let's isolate X

00:58 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Shown below is a parallelogram.

AB = 7

AC = 0.5X

The perimeter of the parallelogram is 21.

Calculate side AC.

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the formula for the perimeter of a parallelogram.
Step 2: Substitute the given values into the perimeter formula.
Step 3: Solve for the unknown variable $X$ .
Step 4: Determine the specific length of AC using the value of $X$ .

Now, let's work through each step:

Step 1: The formula for the perimeter of a parallelogram is given by

P = 2(a + b)

where $a$ and $b$ are the lengths of the two pairs of sides.

Step 2: Substitute the known values into the formula. Here, let $a = AB = 7$ and $b = AC = 0.5X$ .

Perimeter $= 21$ , so:

21 = 2(7 + 0.5X)

Step 3: Solve for the unknown variable $X$ .

First, divide both sides of the equation by 2 to isolate the terms inside the parenthesis:

10.5 = 7 + 0.5X

Subtract 7 from both sides:

3.5 = 0.5X

Multiply both sides by 2 to isolate $X$ :

X = 7

Step 4: Determine the length of AC:

Substitute back to find $0.5X = AC$ :

AC = 0.5 \times 7 = 3.5

Therefore, the length of side AC is $3.5$ .

3

Final Answer

3.5

Key Points to Remember

Essential concepts to master this topic

Perimeter Formula: P = 2(side₁ + side₂) for parallelograms
Substitution: 21 = 2(7 + 0.5X) leads to X = 7
Verification: AC = 0.5(7) = 3.5, so perimeter = 2(7 + 3.5) = 21 ✓

Common Mistakes

Avoid these frequent errors

Using all four sides instead of two pairs
Don't add AB + BC + CD + DA = 7 + 0.5X + 7 + 0.5X! This creates the wrong equation 14 + X = 21, giving X = 7 and AC = 3.5, which seems right but uses flawed logic. Always remember parallelograms have opposite sides equal, so use P = 2(adjacent sides).

Practice Quiz

Test your knowledge with interactive questions

Find the perimeter of the parallelogram using the data below.

FAQ

Everything you need to know about this question

Why don't we add all four sides of the parallelogram?

+

Great question! In a parallelogram, opposite sides are equal. So if AB = 7, then CD = 7 too. If AC = 0.5X, then BD = 0.5X. The formula P = 2(side₁ + side₂) automatically accounts for both pairs!

How do I know which sides are adjacent in the parallelogram?

+

Look at the diagram! Adjacent sides share a common vertex (corner). In this parallelogram, AB and AC share vertex A, so they're adjacent. Use these two different side lengths in your formula.

What if I get a different value for X?

+

Double-check your algebra! Start with $21 = 2(7 + 0.5X)$ , then divide by 2: $10.5 = 7 + 0.5X$ . Subtract 7: $3.5 = 0.5X$ . Finally, divide by 0.5: $X = 7$ .

Can AC ever be longer than AB in this problem?

+

Let's see! Since AC = 0.5X and we found X = 7, we get AC = 3.5. Since AB = 7, we have AB > AC. The coefficient 0.5 ensures AC will always be half of X, making it smaller when X equals AB.

How do I check if my final answer is correct?

+

Substitute back into the perimeter! If AC = 3.5 and AB = 7, then perimeter = 2(7 + 3.5) = 2(10.5) = 21. This matches the given perimeter, so AC = 3.5 is correct! ✓

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